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International Journal of Pure & Applied Mathematical Research

REVIEW OF NUMERICAL STRATEGIES USING FINITE ELEMENT METHODS FOR FLUID DYNAMICS

T. Sivakumar, K. Sulochana

T. Sivakumar, Assistant Professor, Department of Mathematics, Bharathiar University Arts and Science College, Modakkurichi, Erode, Tamil Nadu, India

K. Sulochana, Assistant Professor, Department of Mathematics, Bharathiar University Arts and Science College, Modakkurichi, Erode, Tamil Nadu, India

DOI : https://doi.org/10.5281/zenodo.18383341 Page No : 11-15

Published Online : 2019-06-30

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ABSTRACT

The finite element method (FEM) is a fundamental tool for numerical simulations in fluid dynamics, offering flexible and robust methodologies for modeling complex flow phenomena. This review critically evaluates the primary FEM strategies utilized in computational fluid dynamics (CFD), including classical Galerkin methods, stabilized formulations (SUPG, VMS), least-squares FEM (LSFEM), mixed and hybrid schemes, and adaptive and particle-based techniques (PFEM). The comparative performance, advantages, and limitations of each approach are analyzed, along with representative applications in incompressible, compressible, multiphase, and free-surface flows. Emerging trends such as hp-adaptivity, hybrid formulations, machine learning integration, and high-performance computing implementations are highlighted. This review aims to inform method selection based on flow characteristics, computational requirements, and accuracy needs, underscoring the versatility and evolving capabilities of FEM in contemporary CFD.

Keywords: Fluid dynamics, least squares FEM, particle based FEM , hp adaptivity, hybrid formation

REFERENCES

Babuska, I., & Suri, M. (1994). The p- and hp-versions of the finite element method, basic principles, and properties. SIAM Review, 36(4), 578–632.
Boffi, D., Brezzi, F., & Fortin, M. (2013). Mixed finite element methods and their applications. Springer.
Bochev, P., & Gunzburger, M. (2009). Least-squares finite element method. Springer.
Brooks, A. N., & Hughes, T. J.R. R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection-dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32(1–3), 199–259.
Codina, R. (2000). Stabilization of incompressible Navier–Stokes equations using orthogonal subscales. Computer Methods in Applied Mechanics and Engineering, 190(13–14), 1579–1599.
Donea, J., & Huerta, A. (2003). Finite element methods for flow problems. Wiley.
Elgeti, S., & Sauerland, C. (2015). Finite element methods for incompressible flows: An overview. Archives of Computational Methods in Engineering, 22(3), 389–414.
Hughes, T. J. R. (1995). Multiscale phenomena: Green’s functions, Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles, and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering, 127(1–4), 387–401.
Idelsohn, S. R., Oñate, E., & Del Pin, F. (2004). The particle finite element method: A powerful tool for solving incompressible flows with free surfaces and breaking waves. International Journal for Numerical Methods in Engineering, 61(7), 964–989.
Oñate, E. (1998). Structural analysis with the finite element method: Linear statics, volume 1: Basis and solids. Springer.
Tezduyar, T. E. (1992). Stabilized finite element formulations for incompressible flow computations. Advances in Applied Mechanics, 28, 1–44.
Zienkiewicz, O. C., Taylor, R. L., & Nithiarasu, P. (2013). Finite element method for fluid dynamics (7th ed.). Butterworth-Heinemann.

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REFERENCES

Babuska, I., & Suri, M. (1994). The p- and hp-versions of the finite element method, basic principles, and properties. SIAM Review, 36(4), 578–632.
Boffi, D., Brezzi, F., & Fortin, M. (2013). Mixed finite element methods and their applications. Springer.
Bochev, P., & Gunzburger, M. (2009). Least-squares finite element method. Springer.
Brooks, A. N., & Hughes, T. J.R. R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection-dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32(1–3), 199–259.
Codina, R. (2000). Stabilization of incompressible Navier–Stokes equations using orthogonal subscales. Computer Methods in Applied Mechanics and Engineering, 190(13–14), 1579–1599.
Donea, J., & Huerta, A. (2003). Finite element methods for flow problems. Wiley.
Elgeti, S., & Sauerland, C. (2015). Finite element methods for incompressible flows: An overview. Archives of Computational Methods in Engineering, 22(3), 389–414.
Hughes, T. J. R. (1995). Multiscale phenomena: Green’s functions, Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles, and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering, 127(1–4), 387–401.
Idelsohn, S. R., Oñate, E., & Del Pin, F. (2004). The particle finite element method: A powerful tool for solving incompressible flows with free surfaces and breaking waves. International Journal for Numerical Methods in Engineering, 61(7), 964–989.
Oñate, E. (1998). Structural analysis with the finite element method: Linear statics, volume 1: Basis and solids. Springer.
Tezduyar, T. E. (1992). Stabilized finite element formulations for incompressible flow computations. Advances in Applied Mechanics, 28, 1–44.
Zienkiewicz, O. C., Taylor, R. L., & Nithiarasu, P. (2013). Finite element method for fluid dynamics (7th ed.). Butterworth-Heinemann.

Int. J.of Pure & App Math. Res.

REVIEW OF NUMERICAL STRATEGIES USING FINITE ELEMENT METHODS FOR FLUID DYNAMICS

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